MathDB

P19

Part of 2023 China Team Selection Test

Problems(1)

orz orz orz orz orz Geometry IV, and also orz Haozhe

Source: China TST 2023 Problem 19

3/31/2023
Let A,BA,B be two fixed points on the unit circle ω\omega, satisfying 2<AB<2\sqrt{2} < AB < 2. Let PP be a point that can move on the unit circle, and it can move to anywhere on the unit circle satisfying △ABP\triangle ABP is acute and AP>AB>BPAP>AB>BP. Let HH be the orthocenter of △ABP\triangle ABP and SS be a point on the minor arc APAP satisfying SH=AHSH=AH. Let TT be a point on the minor arc ABAB satisfying TB∣∣APTB || AP. Let ST∩BP=QST\cap BP = Q.
Show that (recall PP varies) the circle with diameter HQHQ passes through a fixed point.
geometry